Twisted line liquids

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Submitted on 1 Jan 1993

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Twisted line liquids

Randall Kamien, T. Lubensky

To cite this version:

Randall Kamien, T. Lubensky. Twisted line liquids. Journal de Physique I, EDP Sciences, 1993, 3

(11), pp.2131-2138. �10.1051/jp1:1993110�. �jpa-00246857�

Classification Physics Abstracts

61.30J 61.25H 05.20

Short Communication

Twisted fine Equids

Randall D. Kamien (~) ^{and T. C.}

Lubensky (~)

(~) School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, ^U-S-A- (~) Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, U-S-A-

(Received

22 July 1993, received in final form 17 August 1993, accepted 30 August 1993)

Abstract We _{propose a} model of directed lines where the average direction has the nature of _a cholesteric liquid crystal. This model, for instance, would describe the liquid of _screw dislocations in the twist-grain-boundary

(TGB)

phase of liquid crystals. We show that the presence of lines does not alter the long wavelength elasticity ^of _a cholesteric and, therefore,

does not stabilize Landau-Peierls instability of the cholesteric phase. We discuss other possible mechanisms for stabilizing the twist-grain-boundary phase.

1. Introduction and _summary.

Directed line

liquids

have received attention

recently

in _a

variety

of

interesting physical

situa- tions.

Composed

of lines which _are

compelled

_{on average} to have tangent vectors

parallel

to _an

imposed

axis,

they

_are central to the

long-wavelength physics

of

high-temperature

_supercon-

ductors in _an external

magnetic field,

both _as

entangled

flux

liquids Ill

and vortex

glass

states

[2].

They

_are also of interest in nematic

polymers

_[3],

electrorheological

fluids and ferrofluids [4]. ^{In this} paper, we will

investigate

a model for chiral line

liquids

in which tangent ^vectors of the lines _are constrained _to rotate in _a helical fashion

along

_some fixed

pitch

axis.

The

entangled

flux

liquid

in _a

superconductor

is

produced

when thermal fluctuations _cause the

regular

Abrikosov vortex lattice to melt. The close

analogy

between the smectic-A

phase

in

liquid crystals

and the

superconducting phase

in metals and that between the nematic-to- smectic-A and

normal-metal-to-superconductor

transition _was

pointed

out _some years ago

by

de Gennes [5]. ^This

analogy

leads to the theoretical

expectation

that _a

liquid crystal analog

of the Abrikosov _vortex

phase

should exist with dislocation lines

replacing

vortex lines. Recent

experiments _[6] confirm that such _a

phase

does exist and that its properties _are identical _to those of the

theoretically predicted

TGB

(twist-grain-boundary) phase iii consisting

of

periodically repeated twist-grain-boundaries,

each composed of

periodically spaced

_screw dislocations whose

axes rotate in _a helical fashion from _one

boundary

to the next. The TGB

phase

_can

presumably

melt to a chiral line

liquid just

_as the Abrikosov

phase

melts _{to a} directed line

liquid.

The chiral line

liquid

we

study

in this _paper is intended to model the melted TGB

phase.

2132 JOURNAL DE PHYSIQUE I N°11

The

entangled

^flux

liquid phase

in _a

superconductor

is in fact the normal-metal

phase,

but with enhanced viscosities.

Polymer nematics,

_on ^the other hand differ from normal

nematici

in that their

splay

elastic constant

Ki

is infinite if all

polymers

extend from _one end of the

sample

to the other [8]. Non-linearities constrained

by

full rotational invariance lead [9] ^to a

renormalized

theory

in which elastic constants take _{on a} momentum

dependence.

In

particular,

the twist and bend Frank constants

diverge

at small momentum while the

splay

modulus

vanishes. It is thus natural to ask whether there _are differences between the chiral line

liquid

and _a cholesteric

liquid crystal.

Our calculations show that the

long-wavelength properties

of the chiral line

liquid

_are identical to those of _a cholesteric

liquid crystal.

A cholesteric is _{a one-}

dimensional

layered

structure with _a

long-wavelength

Landau-Peierls

elasticity

^and associated destruction of

long-range periodic

order

[lo].

The TGB

phase

is also _a

one-dimensional-layered

structure, and

though

its

elasticity

is _more

complex

than that of _a cholesteric

[I II,

its

long-range

order is also fluctuation

destroyed.

A chiral line

liquid

has the _same

long-wavelength

Landau- Peierls

elasticity

_{as a} cholesteric but with elastic constant renormalized

by

the presence of

dislocations. The

splay

elastic constant Ki> which does _not contribute to the Landau-Peierls energy,

however, diverges

_as the _square of _an inverse wavenumber _as it does in _a

polymer

nematic [12].

In this _{paper we} add

explicit degrees

of freedom _to describe the dislocation line

configura-

tions.

Though

the _energy of _a smectic-A _screw dislocation tilted _a small

angle

_{o away} from the director _{goes as} a~

log(I la),

_we will _assume that _{we are} at

sufficiently long

scales that the

tipping

_energy becomes

analytic

[13]. ^{In section 2} we derive _a Landau

theory

for _a

melted,

twisted line

liquid, generalizing

work _on directed line

liquids.

In section 3 _we derive the effec- tive

long-wavelength theory

and find

expressions

for the _new, effective Frank constants of the cholesteric.

Finally

in section 4 _{we comment on a}

possible

_way for the

non-harmonicity

of the

tipping

_energy to stabilize the Landau-Peierls

instability

of the

twist-grain-boundary phase.

2. Derivation of model.

Our system is _a collection of lines described

by

their locations

R;(s)

and

parameterized by ardength

s embedded in and

interacting

with _a cholesteric liquid

crystal

with _a unit director field

n(r).

We model the lines

by

_a local

density

field _p and _a local

tangent-density

vector m.

In terms of the

paths

of the lines

R,(s), P(r)

=

L /ds&~(R«(8) ^{r), (2.1)}

~~~

~~~~ ~i /

_~~

~~)~~&~(R,(s) ^r)

If _our lines do not terminate, then _m must be

divergenceless:

V-m =

~ j

_ds

^l~~j~~ .V&~(Iti(s) r)

=

/ ~ j

_ds

_{)d~(R,(s) r)}

= 0.

~~~~

We will not consider the _case of finite lines because dislocations cannot end within _a

sample.

In the TGB

phase,

_screw dislocations

align parallel

_{on average}

along

the local nematic director _n. We _assume this remains true in the line

liquid phase

and _propose the

following

free energy ^to describe the chiral line

liquid:

F =

/ _d~r

^~

_{(m pan)~}

+

Fn[n], (2.4)

2

~~~~~

j~"

j~j

2

/d~X ^{(Ki (i7.n)~}

^{+ K2}

^{(n.[i7Xnj k0)~}

^+1~3

^(~~i~~i~~j

^{~~ ~~}

is the usual Frank free _energy for _a chiral

liquid crystal

and the constraint i7.m ₌ o is understood. The free _energy favors

(m)

_{= pan.} We could have included _an

anisotropic

tensor

coupling

of the form

~

/

_~~~

_~~" ~°~"~~~~~~" ~°~"~'

_~~'~~

By

rotational invariance I""

= A&"" ₊

Bnpn~.

If _we

keep only

terms up ^to

quadratic

order in the

fields,

_{we are} left

only

with the

isotropic

tensor &"" In

principle,

the

self-coupling

of the vector _can be different

along

the pitch axis of the

twist-grain-boundary phase

than

perpendicular

to

it,

_or in other

words,

if the

pitch points along

the

x-axis,

P~

#

IYY _e I~~.

We

might

also include interactions of the defects with themselves

involving

_m

by

itself. We

believe, however,

that in the

long wavelength

limit, this model captures ^{the essential}

physics.

Additionally,

_starting with the smectic free energy, which involves both the directed field _n and _a

complex

order parameter

#. Following

the

duality mapping

outlined in [14] we can

systematically

derive the action

(2A).

The

ground

state

configuration

^of the

liquid crystal

has _a

spontaneously

broken symmetry.

We choose the

pitch

axis _to lie

along

the x-axis and the

equilibrium

director,

no(r)

=

[o,sin(kox),cos(kox)], (2.7)

to minimize the Frank Free energy. We introduce director fluctuations via _{n = no +} fin. If

we

only

work to

quadratic

order in

fields,

then because _n is _a unit vector, it is sufficient to consider

only

fin with fin _no

= o. Since

(m)

is

parallel

_{to no} in

equilibrium,

_we set

m = pno + t.

(2.8)

Fluctuations of _m

along

_{no are} described

by

_p and fluctuations

perpendicular

to no described

by

t.

We, therefore, require

t _-no

= o. Fluctuations in the

equilibrium

_average of _m will point

along

_no Thus

(t)

₌

o,

and

(m)

₌

(p)no.

The constraint i7.m

= o is _now

no

.i7p

₊ V-t

= o.

(2.9)

and _our harmonic theory becomes

F =

/d~r ~(t _podn)~

+

~dp~)

⁺

^{Fn[n], (2.lo)}

2 2

subject

to no

.Vp

₊ V.t

= o. We have set p = po +

dp

and

F6n[dn]

₌ Fn

[no

+

dn]

=

/ _d~r

[Ki (V. ^fin)

^{~ + K2}

^{(no [Vxdn])}

^{~ + K3}

^(kodnxno

⁺

^{nox[Vxdn] j}

~~'~~~

2

2134 JOURNAL DE PHYSIQUE I N°11

We note that if ko ₌ o then _no

" I and this

theory

reduces to the model derived for directed

line

liquids

in [12].

In order _to

study

the fluctuations in this system _we, seek _a

parameterization

of _m and _p

satisfying

the constraint

(2.9)

and the constraint _no .t ₌ o. The first constraint is satisfied

by setting

P " PO

Poi7.u (2.12a)

t ₌

po(no.i7)u po(u.i7)no, (2.12b)

where _u is _{any vector.} This

equation

is invariant under the transformation _{u -} u'

= u + i7RA.

This _comes about because the constraint

(2.9) only

involves the

longitudinal

_components of

t. There is,

therefore,

_a

gauge-like

symmetry ^of the transverse components. The constraint t. _{no =} o

implies

no'i7(no.u)

= o.

(2,13)

We _can solve this constraint _{on u}

by choosing

_no .u = o. This is tantamount to _a choice of gauge in the _sense that if _{we are}

given

_a u with _{no .u}

#

o _we can, without

making

_u

trivial,

choose _a A such that _no

.(u

₊

i7xA)

= 0. Note that any vector

u(r)

that

depends only

_{on x,}

the component ^of r

along

the

pitch

axis, ^satisfies the constraint

(2,13).

Thus the free _energy

expressed

in term of _u will

depend only

_on

gradients

of _u in the

yz-plane.

3. Rotational

symmetries

and

long-wavelength elasticity.

The free energy for _a chiral line

liquid,

like that for _a

cholesteric,

is invariant with respect to uni- form translations and rotations. It is

precisely

these invariances that lead to the Landau-Peierls

elastic free _energy and the destruction of

long-range

order in _a

one-dimensionally

modulated structure. In order _to obtain the effective

long-wavelength

free energy, ^{it is} useful _to understand how

uniform

translations and rotations _are described

by

the variables _u and fin

describing

de-

viations from the

equilibrium ground

state.

Following

the

analysis

in [15], we write _our fields

u and fin _as

u =

[#, @cos(kox), -@sin(kox)], (3.la)

fin

=

if,

_g

cos(kox),

_-g

sin(kox)], (3.lb)

thus

enforcing

both fin _no

= o and _{u no}

= 0. Because the

ground

state is not

translationally invariant,

it is convenient _to

decompose

_our fields into _{a sum over} Brillouin _zones. We write for _any field X,

x(~)

"

fl xn(~)~~~°~~> (3'2)

n

where

Xn(r) only

has fourier modes with x-components k~ E

[- ~f, ~f].

We _now consider _a

rigid

rotation described

by

the vector fl. Under this rotation _no becomes:

n[

_=no +

[fly cos(kox)

flz

sin(kox),

-fl~

cos(kox),

fl~

sin(kox)]

(fl~z flzy)

_{lo, ko}

cos(kox),

-ko

sin(kox)].

^~~'~~

We _{can now compare}

(3.I)

and

(3.3)

to determine what fields

f

and _{g are}

generated by

uniform

rigid

rotations. We find

f

=

fly cos(koz) flz sin(kox) (3.4a)

g =

-fl~

ko

(fl~z flzy). (3.4b)

If _we write t in terms of

#

and and match the _same

rigid

rotation, _we find

fl~

=

0z# (3.5a)

-flz ₌

0~# (3.5b)

-fl~

ko(fl~z flzy)

=

-ko4

+

sin(kox)0~@

+

cos(kox)0z@ (3.5c)

Then

writing (3.4)

^and

(3.5)

in terms of the

decomposition (3.2) gives (with 2X+

₌ X~ +

X_i

and 2X_

=

-I(Xi X_1))

f+

₌

fl~ (3.6a)

f-

= flz

(3.6b)

go = -fl~ ko

(fl~z flzy) (3.6c)

0~@_ 0z@+ +

ko40

= fl~ +

ko(fl~z flzy) (3.6d)

0~#o

" -flz

(3.6e)

0z#o

₌

fl~ (3.6f)

The free _energy cannot

depend

_on the uniform rotation

angle

fl. It

will, however, depend

_on linear combinations of the fields

#,

_@,

f,

and _g that do _not

depend

_on fl. We _can construct such combinations with the aid of

(3.5)

Note that fl

only depends

_on ~lia the combination 0~@- 0z@+. ^The combination 0~@- + 0z@+ does not generate a

rotation,

and the free _energy

can

depend

_on it.

Finally,

_we note that t will be _nonzero if

0~#o

is _nonzero.

Thus,

_we expect

on quite

general grounds

that F will have the form

a

[(a~go 2kof-)~

₊

(0zgo

₊

2kof+)~j

+ b

[(0y40

+

2f-

_l~ +

(0z40 2f+l~]

F =

/

_d~x

+

c(ko40

_{+ 0~@_ 0z@+ + go)~}

(3.7)

+ d(0~@+ + 0z@_ +

0~#o

_)~

+ e [(°y@+l~ + (°z@+l~ +

(°y@-l~

+

(°z@+l~l

This form is dictated

by

the

underlying

^rotational invariance. In terms of the

original

constants in

(2A)

and

(2.5),

_a

=

(Ki

₊

K3)/2,

and 2b

= c = d

= 2e

=

Ap(.

The values of

b,c,

d and

e are related

by

the

underlying

rotational invariance discussed in section 2. There _are other

terms _as

well,

in addition to

higher

_powers and derivatives of the invariant combinations. We

also note that because _{p =}

-pov

_u,

#n

for _n

#

o will be massive. Likewise, because of the energy of _a

splay configuration fn

will be massive for _n

#

o.

If _we

only

consider the

cholesteric,

then po = 0. In this _{case we are}

only

left with the term

proportional

to a in

(3.7). By integrating

out

fi

_{we see} that

0zg

and

0~g

cannot appear

2136 JOURNAL DE PHYSIQUE I N°11

quadratically.

Thus

they

will first _appear

quartically,

and _we find the classic Landau-Peierls

instability. Including

the

lines,

_we see that _appears without x-derivatives and

always

in combination with other fields.

Upon integrating

out both the terms

proportional

to c and d

disappear.

The

preceding analysis

^showed that _we must

keep

all the modes in the first Brillouin _zone

as well _as

f+

and

f-. Returning

to _our

proposed

model

(2.lo),

_we

integrate

out the modes in the second _{zone, as} well _as #o. ^We

find,

to

leading

order in all

derivatives,

_an effective free

energy for _go1

~g /

_~~~

~)~ 1~2)

(°z90)~

₊ ^~~

)~~~

_~~~~ ~

~~)90j

^~

(3.8)

0

Thus the fluctuation go suffers from the Landau-Peierls

instability.

The

penetration depth

of the smectic, A, is

~

3(K2

₊

3K3)

(3.9)

~

8k( (3K2

+

Ap(I'

and thus the effect of the defects is to decrease the penetration

depth. Though

there is _no

long

_range

order,

sets the scale _over which it is

possible

to _see the

dephasing

of the

ground

state. Consider two defects in the _same

plane

of _constant

phase.

Their mutual

repulsion

forces them apart, ^and as

they

_move

they drag

the

plane

with them. Thus the effect of the

repulsion

should be to make the system "less" ordered.

Additionally,

the free _energy for the

remaining

director mode

fo

is

/

_d~x

^Ki(o~fo)2

+

~~

)

^~~

_j(o~fo)2

₊

(ozfo121

+

(API

+

K~kl) flj

_,

^(3.101

and _so the

fo

fluctuations become massive. In

analogy

with results _on

polymer

nematics [12]

we could interpret part of this _{as a} defect

density dependent divergent

shift in

Ki, namely

K((q)=Ki+@. ~~2 _(3,ll)

qz

4. Stabilization of the

twist-grain-boundary phase.

Because the Landau-Peierls

instability

in the

twist-grain-boundary phase

is due to the under-

lying

rotational

invariance,

it is hard to

imagine

how _any

long-wavelength description

of the

phase

could have

long

_range ^order. As mentioned

above,

the _energy of _a defect tilted at an

angle

a with respect to the

layer

normal is

proportional

to a~

log(1la).

While it is true that upon

integrating

out short-distance modes, the effective _energy will become

analytic

in _a, the

coarse-graining

procedure will be cutoff

by length

scales inherent to the system. ^In particular,

as we

coarse-grain,

the distance between the _screw dislocations decreases. When the distance between them becomes _on the order of the penetration

depth,

_we must stop. ^{If the} screw

dislocations _are close

enough (I.e.

ko is

sufficiently large),

then the defect _energy will remain

non-analytic-

The argument ^{in the} previous section would break down, because _{now we} could have _a term such _as

log(q)t(q).t(-q)

in the free _energy in addition to those in

(2.4). Though again,

_we ^would expect that t would

always

_appear in combination with

fin,

it is hard to _see

how

non-analytic

momentum

dependence

would

change things.

^{Since the} _energy cost of

a bend

in the direction of the cholesteric

pitch

will

pick

_up this

logarithmic

_energy in the

twist-grain- boundary phase,

_we

might consider,

_{as a} toy

model,

_a

slightly

_more

rigid

cholesteric described

by

_a

phase

field _~ with free _energy

F =

/ ) _{il(-q) lB log lilaq~}

lq1~l +

K(ql

+

ql)~l

~1(q).

(4.1)

In this

model,

there is still _a Landau-Peierls

instability,

in that

1~l~

(xi

-

/ A

-B

log

_iaq~

qi~+

K~qj

+

qi

_i~

j ^dq~qidqi

¹

(27r)2 -B

log

[aq~ [q] +

Kq[

/~

(4.2)

j

^dq~

327rli

1IL ^q~

log

_aq~

fi@

167rli

and _so there will still be _a

divergence

with system size, ^{but it} will scale _as the square-root ^of the

logarithm

instead of the

logarithm

itself.

Unless there _are

sufficiently long-ranged

interactions it appears that the

underlying

rota- tional invariance will

always

destabilize the

twist-grain-boundary phase.

5. Conclusions.

We have

investigated

^the structure of _a chiral line

liquid

in order _to determine whether there

were any

qualitative

distinctions between it and _a normal cholesteric. Had there been, this would have

suggested

_{a new}

phase

_or

regime

of smectic

liquid crystals

between the

TGBA phase

and the N*

phase iii.

Our

analysis

shows there will be _no

phase boundary

between the

melted

phase

and the chiral nematic

phase.

While the

qualitative

features of the melted TGB

phase

_are

indistinguishable

from the chiral nematic

phase,

_we expect ^several

quantitative

differences. The presence of

repulsively interacting

defects should increase the

viscosity

of the smectic _as the TGB

phase

is

approached, simply

due _to the

entanglements

of the defects. Our

analysis shows, additionally,

that the

smectic penetration

depth,

will decrease _as the

density

of line defects increases _as described

by (3.9).

Moreover, the presence of the lines increases the stiffness of director fluctuations

along

the

pitch

axis, _as indicated by the

divergence

of Ki

Our

analysis

considered

only

defect lines that _ran from the top ^{to the bottom of the}

sample.

As temperature is increased _we expect dislocation

loops

to form. As

they

do

they

will turn the twisted smectic

phase

into _a pure chiral nematic

phase.

The melted TGB

phase

described here would describe the system

just

above the

crystalline

TGB

phase.

It is

possible

that there

could be _a latent heat associated with the

melting

of the defect

lattice,

just as there

might

be in similar flux

liquid

systems

ill.

2138 JOURNAL DE PHYSIQUE I N°11

Acknowledgements.

It is _a

pleasure

to

acknowledge stimulating

discussions with J. Toner. RDK _was

supported

in part

by

the National Science

Foundation, through

Grant No.

PHY92-45317,

and

through

the Ambrose Monell Foundation. TCL _was

supported

in part

by

the National Science Foundation

through

grants ^No. DMR91-20668 and No. DMR91-22645.

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